Metrization in small and large scale structures

• Autorzy: Miyata T.

Identyfikatory

DOI

10.4064/ba8081-4-2017

• Języki publikacji: EN

Abstrakty

EN

Given a topological structure and a coarse structure on a set, N. Wright gave a necessary and sufficient condition for the set to have a metric inducing simultaneously both the structures. We use the idea of the Alexandroff and Urysohn metrization theorem for topological spaces, to investigate a simultaneous metrization problem for a set with a uniform (and topological) structure and a coarse structure. In particular, we prove that given two metrics d_U and d_C on a set X such that the uniform (topological) structure induced by d_U is compatible in some sense with the coarse structure induced by d_C, there exists a metric d on X which is isometric to d_U in a small scale and to d_C in a large scale. We then apply this idea to show that if, in addition, the uniform space has uniform dimension 0 and the coarse space has asymptotic dimension 0, then there exists an ultrametric d on X which is isometric to d_U in small scale and to d_C in large scale.

Słowa kluczowe

EN

metrization; large scale structure; small scale structure; normal sequence

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2017
• Tom: Vol. 65, no. 1
• Strony: 81--92
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Miyata T.

• Department of Mathematics and Informatics, Graduate School of Human Development and Environment, Kobe University, Kobe, 657-8501 Japan

Bibliografia

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• [6] J. R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence, RI, 1964.
• [7] T. Miyata and Ž. Virk, Dimension-raising maps in a large scale, Fund. Math. 223 (2013), 83–97.
• [8] K. Nagami, Dimension Theory, Academic Press, 1970.
• [9] J. Nagata, On a relation between dimension and metrization, Proc. Japan Acad. 32 (1956), 237–240.
• [10] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser. 31, Amer. Math. Soc., Providence, RI, 2003.
• [11] N. Wright, Simultaneous metrizability of coarse spaces, Proc. Amer. Math. Soc. 139 (2011), 3271–3278.